Two non-conjugate embeddings of $S_3\times {\mathbb{Z}}_2$ into the Cremona group II

We prove more precisely the following main result of \cite{Isk1}. There is two non-conjugate embeddings of $S_3\times {\text{ZZ}}_2$ into the Cremona group that is given by the linear action of this group on the plane and the action on the two-dimensional torus.